If we now replace the scalar field by a massive vector field \(A^{\mu }\) , the most general second-order field equations are called
generalized Proca theories [1]}, [2]}. The application of this kind of theories to cosmology up to
the sixth-order of the vector field self-interactions
in the Lagrangian has been studied in both background and perturbation levels, and compared with the observational data [3]}, [4]}, [5]}, [6]}, [7]}. It has been shown that there exists a de Sitter solution relevant to the late-time expansion in these theories.
In addition, the authors in Ref. [3]} also proposed a dark energy model,
in which the solution always approaches a de Sitter fixed point.
